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I have a Signal of 3 seconds. At different times I start sampling(extract samples) . I need a word that refers to the beginning of each sampling in the signal. I thought of starting point but I am looking for a more accurate term, that sounds more scientific.

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I would suggest:

Sample start

You can see the phrase being used in the context you've mentioned in a discussion on apple.com, they are talking about music, but whatever you are sampling, I am sure the terminology applies equally as well:

The loop needs to start somewhere after the sample start, and the loop needs to end before the sample end.

or to be less idiomatic, you might use:

Sample start point

As used in this video, on lynda.com:

...sample start and end points...

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Consider the term epoch.

Epoch specifically refers to time points or "windows" and is a term commonly used in brain imaging studies. For example, here is how epoch is used to describe the design of one study:

Each exercise run consisted of a 30-s pre-exercise epoch, ten 20-s stimulation epochs separated by ten 20-s control epochs, and a 50-s after-exercise epoch.

For your situation I wouldn't be able to comment without knowing more about what it is you are intending to do, but epoch could be used like so:

The signal was of 3 seconds duration, and sampled into epochs, each 500 ms in length.

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